Prove that the Product of Three Consecutive Positive Integers is Divisible by 6

By BYJU'S Exam Prep

Updated on: September 25th, 2023

In the quest to prove the divisibility of the product of three consecutive positive integers by 6, we explore the fascinating pattern that emerges when we consider these numbers as n, n+1, and n+2. By examining the divisibility properties of consecutive integers, we can demonstrate that their product is divisible by both 2 and 3, two key factors in the divisibility rule for 6.

Table of content

1. Product of Three Consecutive Positive Integers is Divisible by 6
2. What are Consecutive Integers?
3. Prove that the Product of Three Consecutive Positive Integers is Divisible by 6

Product of Three Consecutive Positive Integers is Divisible by 6

If three consecutive numbers n, n + 1, and n + 2 are used, then.

To show that their product is divisible by 6, we need to demonstrate that it is divisible by both 2 and 3.

When a number is divided by 3, the result is always one of the following: 0, 1, or 2.

Let n equal 3p, 3p + 1, or 3p + 2, with p being an integer.

If n = 3p, then n is divisible by 3.

If n = 3p + 1, then n + 2 = 3p + 1 + 2 = 3p + 3 = 3(p + 1) is divisible by 3.

If n = 3p + 2, then n + 1 = 3p + 2 + 1 = 3p + 3 = 3(p + 1) is divisible by 3.

So that n, n + 1 and n + 2 is always divisible by 3.

⇒ n (n + 1) (n + 2) is divisible by 3.

Similar to this, whenever a number is divided by 2, the result is either 0 or 1.

∴ n = 2q or 2q + 1, where q is some integer.

If n = 2q, then n and n + 2 = 2q + 2 = 2(q + 1) are divisible by 2.

If n = 2q + 1, then n + 1 = 2q + 1 + 1 = 2q + 2 = 2 (q + 1) is divisible by 2.

So that n, n + 1 and n + 2 is always divisible by 2.

⇒ n (n + 1) (n + 2) is divisible by 2.

But n (n + 1) (n + 2) is divisible by 2 and 3.

∴ n (n + 1) (n + 2) is divisible by 6.

What are Consecutive Integers?

Consecutive integers are a sequence of numbers that follow each other in order without any gaps. In this sequence, each number is exactly one unit higher than the previous number. For example, the consecutive integers starting from 1 would be 1, 2, 3, 4, 5, and so on. Similarly, the consecutive integers starting from -3 would be -3, -2, -1, 0, 1, 2, and so forth.

Summary: